The expression “financial instrument” is used in a broad sense herein to denote any financial instrument, including but not limited to stocks, bonds, currency, mortgages, futures contracts, and indices.
The term “derivative” is used herein (including in the claims) to denote a financial instrument which derives its value from an underlying financial instrument. An option (e.g., a stock option) is an example of a derivative. Other examples of derivatives are options on currencies and currency futures.
It is well known that the distributions of empirical returns do not follow the log-normal distribution upon which many celebrated results of finance are based. For example, Black F. and Scholes, M., Journal of Political Economy, 81, 637-659, May-June 1973, and Merton R. C., Bell Journal of Economics and Management Science, 4, 141-183, Spring 1973 derive the prices of options and other derivatives of the underlying stock based on such a model. While of great importance and widely used, such theoretical option prices do not match the observed ones. In particular, the Black-Scholes model underestimates the prices of away-from-the-money options, in the sense that the implied volatilities of options of various strike prices form a convex function (sometimes referred to as the “volatility smile”). If the Black-Scholes model were perfect, the implied volatilities would form a flat line as a function of strike price.
There have been several modifications to the standard models in an attempt to correct for these discrepancies, for example those described in Hull, Options, Futures, and other Derivatives, Third Edition, Prentice-Hall, 1997; Eberlein, Keller, and Prause, Journal of Business 71, No. 3, 371-405, 1998; Merton, Journal of Financial Economics, 125-144 (May 1976); Dupire, RISK Magazine, 18-20 (January 1994); Andersen and Andreasen, Review of Derivatives Research, 4, 231-262 (2000); Hull and White, Journal of Finance, 42, 281-300 (1987); and Bouchaud and Potters, Theory of Financial Risks, Cambridge (2000). However, those approaches are complicated or ad-hoc, and do not result in any manageable closed form solution, which is an important result of the Black and Scholes approach.
In contrast, the present invention provides closed form solutions for pricing of derivatives (e.g., European options). The inventive approach is based on a new class of stochastic processes which allow for statistical feedback as a model of the returns of the underlying financial instruments. The distributions of returns implied by these processes closely match those found empirically. In particular they capture features such as the fat tails and peaked middles which are not captured by the standard class of lognormal distributions. The stochastic model employed in the present invention derives from a class of processes, described in Borland L., Phys. Rev. E 57, 6634, 1998, that have been recently developed within the framework of statistical physics, namely within the active field of Tsallis nonextensive thermostatistics (aspects of which are described at Tsallis C., J. Stat. Phys. 52, 479, 1988, and Curado E. M. F. and Tsallis C., J. Phys. A 24 L69, 1991; 24, 3187, 1991; and 25 1019, 1992).
The stochastic processes employed to model stock returns in accordance with the present invention can be interpreted as if the driving noise follows a generalized Wiener process governed by a fat-tailed Tsallis distribution of index q>1. For q=1, the Tsallis distribution coincides with a Gaussian and the standard stock-price model is recovered. However, for q>1 these distributions exhibit fat tails and appear to be good models of real data, as shown in FIG. 1. In FIG. 1, the empirical distribution of the log daily price returns (ignoring dividends and non-trading days) to the demeaned Standard & Poor's 500 index is plotted. Returns were normalized by the sample standard deviation of the series which is 19.86% annualized, and then binned. For comparison, the solid curve superimposed on the data is a Tsallis distribution of index q=1.43. It is clear from FIG. 1 that the Tsallis distribution provides a much better fit to the empirical distribution than a lognormal distribution (the dashed curve also shown).
Another example is shown in FIG. 2, which plots the distribution of high frequency log returns for 10 NASDAQ high-volume stocks. The data points are generated as in FIG. 1, but indicate returns at 1 minute intervals rather than daily intervals. As in FIG. 1, returns are normalized by the sample standard deviation. The solid curve superimposed on the empirical data is a Tsallis distribution of index q=1.43 which provides a very good fit to the data. Stock return movements according to the stochastic processes employed in accordance with the present invention have been simulated, and the inventor has found that the distributions of returns over varying time-lags match corresponding empirical observations very well.
The standard model for stock price movement is thatS(t+τ)=S(τ)eY(t),  (1)where t is a time delay, time interval, or timescale, and where Y follows the stochastic processdY=μdt+σdω  (2)across t, the drift, μ, is the mean rate of return of the stock, and σ2 is the variance of the logarithmic return of the stock. We can set τ=0 without loss of generality, and do so below. The driving noise ω is a Brownian motion defined with respect to a probability measure F. It represents a Wiener process and has the propertyEF[dω(t)dω(t′)]=dtdt′δ(t−t′)  (3)where the notation EF[ ] means the expectation value with respect to the measure F. Note that the conditional probability distribution of the variable ω satisfies the Fokker-Planck equation
                                                                        ∂                P                            ⁢                                                          ⁢                              (                                  ω                  ,                                      t                    |                                          ω                      ′                                                        ,                                      t                    ′                                                  )                                                    ∂              t                                =                                    1              2                        ⁢                                          ∂                                                     2                                                            ∂                                  ω                  2                                                      ⁢            P            ⁢                                                  ⁢                          (                              ω                ,                                  t                  |                                      ω                    ′                                                  ,                                  t                  ′                                            )                                      )                            (        4        )            and is distributed according to
                              P          ⁢                                          ⁢                      (                                          ω                ⁡                                  (                  t                  )                                            ,                              t                |                                  ω                  ⁢                                                                          ⁢                                      (                                          t                      ′                                        )                                                              ,                              t                ′                                      )                          =                              1                                          2                ⁢                                                                  ⁢                π                ⁢                                                                  ⁢                                  (                                      t                    -                                          t                      ′                                                        )                                                              ⁢                                          ⁢          exp          ⁢                                          ⁢                      (                          -                                                                    (                                                                  ω                        ⁢                                                                                                  ⁢                                                  (                          t                          )                                                                    -                                              ω                        ⁢                                                                                                  ⁢                                                  (                                                      t                            ′                                                    )                                                                                      )                                    2                                                  2                  ⁢                                                                          ⁢                                      (                                          t                      -                                              t                        ′                                                              )                                                                                                          (        5        )            In addition one chooses t′=t0 and ω(t0)=0 that this defines a Wiener process, which is distributed according to a zero-mean Gaussian. It is well known that this model gives a normal distribution with drift μ(t−t0) and variance σ2(t−t0) for the variable Y. This can be seen for example by rewriting Equation (2) as
                              ⅆ                                          ⁢                      (                                          Y                -                                  μ                  ⁢                                                                          ⁢                  t                                            σ                        )                          =                  ⅆ          ω                                    (        6        )            which indicates that we can substituteω=(Y−μt)/σ  (7)into Eq (5). This obtains the well-known log-normal distribution for the stock returns over timescale T, after inserting Y=ln S(T)/S(0):
                              P          ⁢                                          ⁢                      (                                          ln                ⁢                                                                  ⁢                S                ⁢                                                                  ⁢                                  (                  T                  )                                            ,                              T                |                                  ln                  ⁢                                                                          ⁢                  S                  ⁢                                                                          ⁢                                      (                    0                    )                                                              ,              0                        )                          =                  N          ⁢                                          ⁢          exp          ⁢                                          ⁢                      (                          -                                                                    ln                    ⁢                                                                                  ⁢                                                                  S                        ⁢                                                                                                  ⁢                                                  (                          T                          )                                                                                            S                        ⁢                                                                                                  ⁢                                                  (                          0                          )                                                                                                      -                                      μ                    ⁢                                                                                  ⁢                                                                  (                        T                        )                                            2                                                                                        2                  ⁢                                                                          ⁢                                      σ                    2                                    ⁢                                                                          ⁢                                      (                    T                    )                                                                        )                                              (        8        )            Based on this stock-price model, Black and Scholes were able to establish a pricing model to obtain the fair value of options on the underlying stock whose price is S. The present invention provides a new derivative pricing model based on a new class of stochastic processes that, until the present invention, had not been used for derivative pricing.